No, that's the beauty of equilibrium - for a given reaction at equilibrium reaction quotient is always identical and equals Keq. When reaction quotient is not equal to Keq, you are not at equilibrium and system will react till it gets to Keq. How fast it will get there depends on the reaction kinetics, so sometimes it is almost impossible to observe system with Q <> Keq (for example neutralization reactions are very fast), sometimes reaction is slow and after mixing Q<>Keq.